Determining embedding dimension for phase-space reconstruction using a geometrical construction

TLDR

Noise obscures the precise determination of the embedding dimension d_E. The study aims to determine an acceptable minimum embedding dimension by examining how near‑neighbor relationships change when the embedding dimension increases from d to d+1. The authors introduce a criterion that tracks near‑neighbor behavior across successive dimensions, assess how noise alters this criterion, and quantify the error incurred when selecting a dimension smaller than the true d_E. They find that the attractor is fully unfolded when no new nearest neighbors appear in dimension d_E, and that this criterion can aid practical analysis of observed time series.

Abstract

We examine the issue of determining an acceptable minimum embedding dimension by looking at the behavior of near neighbors under changes in the embedding dimension from d\ensuremath{\rightarrow}d+1. When the number of nearest neighbors arising through projection is zero in dimension ${\mathit{d}}_{\mathit{E}}$, the attractor has been unfolded in this dimension. The precise determination of ${\mathit{d}}_{\mathit{E}}$ is clouded by ``noise,'' and we examine the manner in which noise changes the determination of ${\mathit{d}}_{\mathit{E}}$. Our criterion also indicates the error one makes by choosing an embedding dimension smaller than ${\mathit{d}}_{\mathit{E}}$. This knowledge may be useful in the practical analysis of observed time series.